205 research outputs found
Another integrable case in the Lorenz model
A scaling invariance in the Lorenz model allows one to consider the usually
discarded case sigma=0. We integrate it with the third Painlev\'e function.Comment: 3 pages, no figure, to appear in J. Phys.
Can the Benjamin-Feir instability spawn a rogue wave?
Abstract. Recent work by our research group has shown that wave damping can have a surprisingly strong effect on the evolution of waves in deep water, even when the damping is weak. Whether damping is or is not included in a theoretical model can change the outcome in terms of both stability of wave patterns and frequency downshifting. It is conceivable that it might affect the early development of rogue waves as well
A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy
In the present paper we begin studies on the large time asymptotic behavior
for solutions of the Cauchy problem for the Novikov--Veselov equation (an
analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are
focused on a family of reflectionless (transparent) potentials parameterized by
a function of two variables. In particular, we show that there are no isolated
soliton type waves in the large time asymptotics for these solutions in
contrast with well-known large time asymptotics for solutions of the KdV
equation with reflectionless initial data
Kink Dynamics in a Topological Phi^4 Lattice
It was recently proposed a novel discretization for nonlinear Klein-Gordon
field theories in which the resulting lattice preserves the topological
(Bogomol'nyi) lower bound on the kink energy and, as a consequence, has no
Peierls-Nabarro barrier even for large spatial discretizations (h~1.0). It was
then suggested that these ``topological discrete systems'' are a natural choice
for the numerical study of continuum kink dynamics. Giving particular emphasis
to the phi^4 theory, we numerically investigate kink-antikink scattering and
breather formation in these topological lattices. Our results indicate that,
even though these systems are quite accurate for studying free kinks in coarse
lattices, for legitimate dynamical kink problems the accuracy is rather
restricted to fine lattices (h~0.1). We suggest that this fact is related to
the breaking of the Bogomol'nyi bound during the kink-antikink interaction,
where the field profile loses its static property as required by the
Bogomol'nyi argument. We conclude, therefore, that these lattices are not
suitable for the study of more general kink dynamics, since a standard
discretization is simpler and has effectively the same accuracy for such
resolutions.Comment: RevTeX, 4 pages, 4 figures; Revised version, accepted to Physical
Review E (Brief Reports
Some Recent Developments on Kink Collisions and Related Topics
We review recent works on modeling of dynamics of kinks in 1+1 dimensional
theory and other related models, like sine-Gordon model or
theory. We discuss how the spectral structure of small perturbations can affect
the dynamics of non-perturbative states, such as kinks or oscillons. We
describe different mechanisms, which may lead to the occurrence of the resonant
structure in the kink-antikink collisions. We explain the origin of the
radiation pressure mechanism, in particular, the appearance of the negative
radiation pressure in the and models. We also show that the
process of production of the kink-antikink pairs, induced by radiation is
chaotic.Comment: 26 pages, 9 figures; invited chapter to "A dynamical perspective on
the {\phi}4 model: Past, present and future", Eds. P.G. Kevrekidis and J.
Cuevas-Maraver; Springer book class with svmult.cls include
Leading Order Temporal Asymptotics of the Modified Non-Linear Schrodinger Equation: Solitonless Sector
Using the matrix Riemann-Hilbert factorisation approach for non-linear
evolution equations (NLEEs) integrable in the sense of the inverse scattering
method, we obtain, in the solitonless sector, the leading-order asymptotics as
tends to plus and minus infinity of the solution to the Cauchy
initial-value problem for the modified non-linear Schrodinger equation: also
obtained are analogous results for two gauge-equivalent NLEEs; in particular,
the derivative non-linear Schrodinger equation.Comment: 29 pages, 5 figures, LaTeX, revised version of the original
submission, to be published in Inverse Problem
Surface polaritons on left-handed cylinders: A complex angular momentum analysis
We consider the scattering of electromagnetic waves by a left-handed cylinder
-- i.e., by a cylinder fabricated from a left-handed material -- in the
framework of complex angular momentum techniques. We discuss both the TE and TM
theories. We emphasize more particularly the resonant aspects of the problem
linked to the existence of surface polaritons. We prove that the long-lived
resonant modes can be classified into distinct families, each family being
generated by one surface polariton propagating close to the cylinder surface
and we physically describe all the surface polaritons by providing, for each
one, its dispersion relation and its damping. This can be realized by noting
that each surface polariton corresponds to a particular Regge pole of the
matrix of the cylinder. Moreover, for both polarizations, we find that there
exists a particular surface polariton which corresponds, in the large-radius
limit, to the surface polariton which is supported by the plane interface.
There exists also an infinite family of surface polaritons of
whispering-gallery type which have no analogs in the plane interface case and
which are specific to left-handed materials.Comment: published version. v3: reference list correcte
Discrete breathers in classical spin lattices
Discrete breathers (nonlinear localised modes) have been shown to exist in
various nonlinear Hamiltonian lattice systems. In the present paper we study
the dynamics of classical spins interacting via Heisenberg exchange on spatial
-dimensional lattices (with and without the presence of single-ion
anisotropy). We show that discrete breathers exist for cases when the continuum
theory does not allow for their presence (easy-axis ferromagnets with
anisotropic exchange and easy-plane ferromagnets). We prove the existence of
localised excitations using the implicit function theorem and obtain necessary
conditions for their existence. The most interesting case is the easy-plane one
which yields excitations with locally tilted magnetisation. There is no
continuum analogue for such a solution and there exists an energy threshold for
it, which we have estimated analytically. We support our analytical results
with numerical high-precision computations, including also a stability analysis
for the excitations.Comment: 15 pages, 12 figure
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